Quantum Field Theory Chap 7 Renormalization

Ling-Fong Li December 21, 2014

Contents

1 Renormalization 1 1.1 Renormalization in λφ4 Theory ...... 2 1.1.1 Mass and wavefunction renormalization ...... 4 1.1.2 Coupling constant renormalization ...... 6 1.2 BPH renormalization ...... 8 1.2.1 Power counting ...... 9 1.2.2 Analysis of divergences ...... 10 1.3 Regularization ...... 13 1.3.1 Pauli-Villars Regularization ...... 13 1.3.2 Dimensional regularization ...... 18

2 Power counting and Renormalizability 20 2.1 Theories with fermions and scalar fields ...... 20 2.2 Theories with vector fields ...... 24 2.3 Composite operator ...... 27

3 Renormalization group 29 3.1 Renormalization group equation 29 3.2 Effective coupling constant ...... 31

4 Appendix n-dimensional Integration 34 4.1 n-dimensional “spherical”coordinates ...... 34 4.1.1 (a) n = 2 ...... 34 4.1.2 (b) n = 3 ...... 34 4.1.3 (c) n = 4 ...... 35 4.1.4 (d) General n ...... 36 4.2 Some integrals in dimensional regularization ...... 36

1 Renormalization

Many people who have studied relativistic quantum field theory find the most diffi cult part is the theory of renormalization. The relativistic field theory is full of infinities which need to be taken care of before the theoretical predictions can be compared with experimental measurements. These infinities look formidable at first sight. It is remarkable that over the years consistent ways have been found to make sense of these apparently divergent theories and the results compared successfully with experiments. The theory of renormalization is a prescription which consistently isolates and removes all these infinities from the physically measurable quantities. Note that the need for renomalization is quite general and is not unique to the relativistic field theory. For example, consider an electron moving inside a solid. If the interaction between electron and the lattice of the solid is weak enough, we can use an effective mass m∗ to describe its response to an externally applied force and this effective mass is certainly different from the mass m measured outside the solid where there is no interaction. Thus the electron mass is changed (renormalized) from m to m∗ by the interaction of the electron with the lattice in the solid. In this simple case, both m and m∗ are measurable and hence both are finite. For the relativistic field theory, the situation is the same except for two important differences. First the renormalization due to the interaction is generally infinite (corresponding to the divergent loop diagrams). These infinities, coming from the contribution of high momentum modes are present even for the cases where the interactions are weak. Second, there is no way to switch off the interaction between particles and the quantities in the absence of interaction, bare quantities, are not measurable. Roughly speaking, the program of removing the infinities from physically measurable quantities in relativistic field theory, the renormalization program, involves shuffl ing all the divergences into bare quantities which are not measurable. In other words, we can redefine the unmeasurable quantities to absorb the divergences so that the physically measurable quantities are finite. The renormalized mass which is now finite can only be determined from experimental measurement and can not be predicted from the theory alone. Eventhough the concept of renormalization is quite simple, the actual procedure for carrying out the operation is quite complicated and intimidating. In this chapter, we will give a bare bone of this pro- gram. Note that we need to use some regularization procedure ([?]) to make these divergent quantities finite before we can do mathematically meaningful manipulations. It turns out that not every relativistic field theory will have this property that all divergences can be absorbed into redefinition of few physical parameters. Those which have this property are called renormalizable theories and those which don’t are called unrenormalizable theories. This has became an important criteria for choosing a right theory because we do not really know how to handle the unrenormalizable theory.

1.1 Renormalization in λφ4 Theory To avoid complicaiton from spin and gauge dependence, we consider a simple example of λφ4 theory where the Lagrangian is given by, = 0 + I L L L 1 2 2 2 λ0 4 0 = [(∂µφ ) µ φ ] , I = φ L 2 0 − 0 0 L − 4! 0 2 Here µ0 , λ0, φ0 are usually called the bare or unrenormalized quantities and will be renormalized by ♠the interactions. Feynman rule♠ The vertex and propagator are given by

Fig 1 Feynman rule for λϕ4 theory

2 Here p is the momentum carried by the line and µ is the bare mass term in 0. 0 L (a) 4-momentum conservation at each vertex.

(b) Integrate over internal momenta not fixed by momentum conservation

(c) For the discussion of renormalization, we do not give propagators for external lines

Simple example First consider the two point function (propagator) is defined by ♠ ♠ 4 ip x i∆ (p) = d xe− · 0 T (ϕ (x) ϕ (0)) 0 h | 0 0 | i Z and has contribution from following graphs for example, We define the term one-particle-irreducible, or 1PI as those graphs which can not be made disconnected by cutting any one internal line. It is clear that we can express a general graph in terms of some collections of 1PI graphs connected by a single line. For expample in the above graphs diagram (a) is an 1PI graph while diagram (b) is not. It is not hard to see that we can attribute all the divergence to the 1PI subgraphs. For example, in daigram (b) the divergences are due to the 1PI in 1-loop. So we can concentrate on 1PI graphs for the discussion of renormalization because in the one particle reducible graph the internal line which can be cut to make the graphs a disconnected♠ one will not be integrated over. It is easy to see that we can write the full propagator in terms of 1PI self energy♠ graph as a geometric series,

i i i i∆ (p) = + iΣ p2 + (1) p2 µ2 + iε p2 µ2 + iε − p2 µ2 + iε ··· − 0 − 0 − 0 i

= p2 µ2 Σ(p2) + iε − 0 − Here Σ p2 contains all IPI self energy graphs. In one-loop we have following divergent 1PI graphs,
♠ ♠

The 1-loop self energy graph given below,

iλ d4l i iΣ(p) = 0 − − 2 (2π)4 l2 µ2 + iε Z − 0 is quadratically divergent. The 1-loop 4-point function are with graph (a) gives the contribution

(iλ )2 d4l i i Γ(p2) = 0 2 (2π)4 (l p)2 µ2 + iε l2 µ2 + iε Z − − 0 − 0 and is logarithmically divergent. Note that in Γ p2 , inside the integral the dependence on the external momentum p is in the combination (p l) in the denominator. This means that if we differentiate Γ p2
with respect to the external momenta p−, power of l will increase in denominator and will make the integral
more convergent,

2 4 ∂ 2 1 ∂ 2 λ0 d l (l p) p 1 2 Γ(p ) = 2 pµ Γ(p ) = 2 4 2− ·2 2 2 2 convergent ∂p 2p ∂pµ p (2π) [(l p) µ + iε] l µ + iε −→ Z − − 0 − 0 This is true whenever the external momenta ocurrs in company with internal momenta in the denominator. Thus if we expand Γ(p2) in Taylor series,

2 2 Γ(p ) = a0 + a1p + ...

the coeffi cients are all derivatives of Γ(p2) with respect to p and the divergences are contained in first few terms in the expansion. So we will use Taylor expansion to separate the divergent quantities from the convergent terms. In our simple case, if we write

Γ(p2) = Γ(0) + Γ(˜ p2)

then then only the first term Γ(0) is divergent and second term Γ(˜ p2) which contains all higher derivatives terms is finite. Other 1-loop graphs are either finite or contain the above graphs as subgraphs.

1.1.1 Mass and wavefunction renormalization In 1PI self energy, the expansion in external momentum p will have 2 divergent terms because it is quadrat- ically divergent, 2 2 2 2 2 2 2 Σ(p ) = Σ(µ ) + (p µ )Σ0 (µ ) + Σ(˜ p ) (2) − where µ2 is an arbitrary reference point for the Taylor expansion. It is clear that the first term Σ(µ2) is 2 2 ˜ 2 quadratically and the second term Σ0 (µ ) logarithmically divergnet. The 3rd term Σ(p ) is finite and satisfies the conditions, 2 2 2 Σ(˜ µ ) = 0, Σ˜ 0 (µ ) = 0 Complete propagator is then i i∆(p2) = p2 µ2 Σ(µ2) (p2 µ2)Σ 2 Σ(˜ p2) − 0 − − − 0 − Suppose we choose µ2 such that

µ2 Σ(µ2) = µ2 mass renormalization (3) 0 − then ∆(p2) will have a pole at p2 = µ2 . The position of the pole µ2 can be interpreted as physical mass 2 2 and µ0 is the bare mass which appears in the original Lagrangian. Note that in Eq ( 3) Σ(µ ) is divergent. 2 2 2 Thus we require the bare mass µ0 to be divergent in such a way that the combination µ0 Σ(µ ), the 2 − physical mass, is finite. In other words, we blame the divergence on the bare mass µ0, which is not a measurable quantity and♠ diverges in such a way that the combination with the term from the interaction Σ(µ2) is finite. This interpretation looks strange and is hard to accept at first sight but it is logically consistent and workable Using the mass renormalization♠ condition in Eq (3) we can write the full propagator as i i∆(p2) = (p2 µ2)[1 Σ 2 (µ2)] Σ(˜ p2) − − 0 − 2 ˜ 2 Since Σ0(µ ) and Σ(p ) are both of order λ0 or higher, we can approximate

2 2 2 2 2 Σ(˜ p ) (1 Σ0 µ )Σ(˜ p ) + O λ → − 0 2 2 ˜ 2
2
Note that the term we add Σ0 µ Σ(p ) is of order λ0 and is negligible in the perturbative expansion in♠ λ . The purpose for doing− this is to be able to factor out the combination (1 Σ 2 µ2 ) to wrtie the 0
0 full propagator as − ♠
2 iZφ 1 2 2 i∆(p ) = with Z = 1 + Σ0 (µ ) 2 2 2 φ 2 2 p µ Σ(˜ p ) + iε 1 Σ0 (µ ) ≈ − − − where we have expand the denominator as

1 2 2 2 1 + Σ0 (µ ) + O λ 1 Σ 2(µ2) ≈ 0 − 0
Now the divergence is aggravited into a multiplicative factor Zφ and can be removed by rescaling the field, 1 φ = φ0 Zφ so that the propagator for φ is p

4 ipx 1 2 i i∆R(p) = d xe− 0 T (φ(x)φ(0)) 0 = i∆(p ) = h | | i Zφ p2 µ2 Σ(˜ p2) + iε Z − − which is now completely finite. Zφ is usually called the wave function renormalization constant. The new field φ is called the renormalized field operator. Again we shuffl e the divergence into the bare or ♠ unrenormalized field φ0 so that the renormalized field φ will have finite matrix elements. Note that this method of removing the first few terms in the Taylor expansion about♠ certain point is sometime called the substration scheme. In our case we expand the self energy Σ(p2) around the point µ2 as in Eq (2) and subtract the first two terms in the expansion, Σ2(µ2), and p2 µ2 Σ 2(µ2). The left − 0 over Σ(˜ p2) will satifies the conditions
∂Σ(˜ p2) Σ(˜ p2) = 0, = 0 (4) p2=µ2 ∂p2 2 2 p =µ

In fact, the conditions in Eq (4) contain all the information about the expansion points and terms to be subtracted. Very often these conditions sometimes are used to specify the normalization procedure. Note also that since our purpose is to remove the divergence we do not have to choose the expansion 2 2 point µ to be the physical mass as we have done. Suppose we use some other point µ1 for the Taylor expansion, 2 2 2 2 2 2 2 Σ(p ) = Σ(µ ) + (p µ )Σ0 (µ ) + Σ(˜ p ) 1 − 1 1 Then the full propagator is i i∆(p2) = p2 µ2 Σ(µ2) (p2 µ2)Σ 2(µ2) Σ(˜ p2) − 0 − 1 − − 1 0 1 − We can still define the physical mass µ2 to be the value of p2 where the propagator i∆(p2) has a pole, i.e.

2 2 2 2 2 2 2 2 µ µ Σ(µ ) (µ µ )Σ0 (µ ) Σ(˜ µ ) = 0 − 0 − 1 − − 1 1 − 2 2 2 2 2 This will give the physical mass µ as a complicate function of bare mass µ0 divergent quantities, Σ(µ1), Σ0 (µ1) ˜ 2 2 ♠ and convergent one Σ(µ ). The parameter µ1 is now a parameter of the theory and all the physical quan- tities are then functions of this parameter. Its value needs to be determined by comparing the theoretical calculation of certain physical quantity with the experimental measurement. This might looks cubersome, but is workable in principle. For more general Green’sfunctions♠ of the renomalized fields, we define the renormalized Green’sfunc- tions in terms of renormalized fields,

(n) G (x1 . . . xn) = 0 T (φ(x1) φ(xn)) 0 R h | ··· | i n/2 n/2 (n) = Z− 0 T (φ (x1) φ (xn)) 0 = Z− G (x1 . . . xn) φ h | 0 ··· 0 | i φ 0 so that the renormalized Green’sfunction will be free of divergences. ♠ In summary, the mass and wave function renormalization remove all♠ the divergence in 1PI self energy graphs by redefining the mass and field operator. In other words, all the divergences have been shuffl ed 2 4 into unobserable bare mass µ0 and bare field operator φ0. The program of renormalization in λφ theory is to show that this procedure can be extended to higher order consistently. This is a daunting task and will not be discussed here. Later we will illustrate with counter terms scheme to make it more plausible.

1.1.2 Coupling constant renormalization

(4) For 1PI 4-point functions Γ (p1 p4), there are 3 1-loop diagrams, ···

If we include the lowest order tree diagram, we have

(4) Γ (s, t, u) = iλ0 + Γ(s) + Γ(t) + Γ(u) 0 − where 2 2 2 2 s = (p1 + p2) , t = (p1 p3) , u = (p1 p4) , s + t + u = 4µ − − are the standard Mandelstam variables. Since these contributions are logarithmically divergent, we need to isolate only one term in the Taylor expansion, or one substraction, to make this finite. To remove this divergence from the 4-point function, we need to make the substraction of 4-point function at some 4µ2 kinematical point. For conveience we can choose the symmetric point s = t = u = , 0 0 0 3 (4) Γ (s, t, u) = iλ0 + 3Γ(s0) + Γ(˜ s) + Γ(˜ t) + Γ(˜ u) 0 − where Γ(˜ s) = Γ(s) Γ(s0), − is finite. We now combine the divergent terms and define the coupling constant renormalization constant Zλ by 1 iλ0 + 3Γ(s0) = iZ− λ0 − − λ Thus (4) 1 Γ (s, t, u) = iZ− λ0 + Γ(˜ s) + Γ(˜ t) + Γ(˜ u) 0 − λ At the symmetric point we see that

(4) 1 Γ (s0, t0, u0) = iZ− λ0 0 − λ (4) with Γ(˜ s0) = Γ(˜ t0) = Γ(˜ u0) = 0. The renormalized 1PI 4 point function Γ is related to Green’s function by 4 (4) 1 (4) ΓR = [i∆R(pj)]− GR j=1 Y which implies the following relation between renormalized and un-renormalized 1PI 4-point functions,

(4) 2 (4) ΓR (s, t, u) = ZφΓ0 (s, t, u) Or (4) 2 1 Γ (s, t, u) = Z [ iZ− λ0 + Γ(˜ s) + Γ(˜ t) + Γ(˜ u)] R φ − λ Here we see that the divergences are contained in the first term, so we define the renormalized coupling constant λ by 2 1 λ = ZφZλ− λ0 (5)

In other words, we let the bare or unrenormalized coupling constant λ0 be divergent in such a way that 2 1 the combination ZφZλ− λ0 is finite. The renormalized 1PI 4-point function is then

(4) 2 (4) 1 2 2 2 Γ (p1, , p4 ) = Z Γ = iZ− Z λ0 + Z [Γ(˜ s) + Γ(˜ t) + Γ(˜ u)] = iλ + Z [Γ(˜ s) + Γ(˜ t) + Γ(˜ u)] R ··· φ 0 − λ ϕ ϕ − ϕ ˜ 2 2 Since Zϕ = 1 + O(λ0), Γ = O(λ0) λ = λ0 + O(λ0), we can approximate

(4) 3 Γ ( p1, , p4) = iλ + Γ(˜ s) + Γ(˜ t) + Γ(˜ u) + O(λ ) R ··· − which is completely finite. In Eq (5), we see that the unrenormalized coupling constant λ0 has to diverge in such a way that the (4) renomalized coupling constant λ is finite. Note that λ is measured in the physical processes involving ΓR , i.e. 2 (4) 4µ Γ ( p1, , p4) = iλ, at symmetric point s0 = t0 = u0 = R ··· − 3 Here we will illustrate the fact different normalization schemes will give the same physical result. Suppose ♠ 2 we make the Taylor expansion about some other point, s = s1, t = t1, u = u1, with s1 + t1 + u1 = 4µ . Then it is not hard to see that the renomalized 4 point function is of the form

(4) 3 Γ0 ( p1, , p4) = iλ0 + Γ˜0(s) + Γ˜0(t) + Γ˜0(u) + O(λ0 ) R ··· − where Γ˜0(s) = Γ(s) Γ(s1), Γ˜0(t) = Γ(t) Γ(t1), Γ˜0(u) = Γ(u) Γ(u1) − − − and 1 2 1 iλ0 + Γ(s1) + Γ(t1) + Γ(u1) = iZ0− λ0, λ = Z Z0− λ0 − − λ φ λ (4) (4) (4) Note that ΓR and ΓR0 are obtained from Γ0 by substracting out some constants. So if we consider the (4) (4) angular distribution in the scattering reaction, these two functions ΓR , ΓR0 will have the same shape and can only differ by a constant. Suppose we write the angular distribution as (4) (4) Γ ( p1, , p4) = iλ + f1 (θ) , Γ0 ( p1, , p4) = iλ0 + f1 (θ) + b R ··· − R ··· − (4) (4) where b is some contant. Since we need to compare the computed renomalized amplitudes ΓR , ΓR0 with some experimental measurement, say at θ = θ0, in order to determine the coupling contants λ and λ0, we get iλ + f1 (θ0) = iλ0 + f1 (θ0) + b, or iλ = iλ0 + b − − − − (4) (4) This shows that these two amplitudes ΓR , ΓR0 from 2 different renormalization schemes are the same. After the mass, wave function and coupling constant renormalization, we can rewrite the original♠ Lagrangian (unrenormalized Lagrangian)

1 2 2 2 λ0 4 0 = [(∂µφ ) µ φ ] φ L 2 0 − 0 0 − 4! in terms of renormalized quantities and counterterms

0 = + ∆ (6) L L L where 1 2 2 2 λ 4 = [(∂µφ) µ φ ] φ L 2 − − 4! is called the renormalized Lagrangian and 2 1 2 2 2 δµ 2 λ(Zλ 1) 4 ∆ = 0 = (Zφ 1)[(∂µφ) µ φ ] + φ − − φ L L − L 2 − − 2 − 4! are called the counterterms. Here we have used the relations,

1 2 2 2 − 2 1 2 µ = δµ + µ0 , φ = Zφ φ0 , λ = Zλ− Zφλ0

1.2 BPH renormalization An equivalent, perhaps more organized way of carrying out renormalization is the BPH (Bogoliubov, Parasiuk and Hepp) renormalization scheme. The essential idea here is to use the counter terms Lagrangian ∆ as a device to cancel the divergences. In this scheme, the relation between lower divergences and high orderL one is more transparent. This scheme also provides a simple criterion for deciding whether a given theory is renormalizable or not. We will simply illustrate this procedure and discuss some simple points. (a) Starts with renormalized Lagrangian 2 1 2 µ 2 λ 4 = (∂µφ) φ φ L 2 − 2 − 4! Generate free propagator and vertices from this Lagrangian. (b) The divergent parts of one-loop 1PI diagrams are isolated by Taylor expansion. Construct a set of counter terms ∆ (1) to cancel these divergences. L (c) A new Lagrangian (1) = + ∆ (1) is used to generate 2-loop diagrams and counter terms ∆ (2) to cancel 2-loops divergences.L L ThisL sequence of operation is iteratively applied. L To illustrate the usefulness of BPH scheme, we need to make use of the following power counting method. 1.2.1 Power counting Superficial degree of divergence D is defined as D = (] of loop momenta in numerator) (] of loop momenta in denominator) − We define the following quantities, B= number of external lines IB= number of internal lines n= number of vertices Counting all the lines in the graph, we get 4n = 2(IB) + B When we compute the Feynman diagrams we need to integrate over the momentum of each internal line, subjected to the 4-momentum conseravation at each vertex But there is an overall 4-momentum conseravation which do not depend on the internal momentum. Then the number of loops, or unrestricted intenal momenta, L is given by L = IB n + 1 − The superficial degree of divergence is D = 4L 2(IB) − Eliminating n, L and (IB), we get D = 4 B − Thus D < 0 for B > 4.Note that in this case D is independent of n, the order of pertubation, and depends only on B, the number of external lines. This means that all the superficial divergences reside in some small number of Green’s functions. The λφ4 theory has the symmetry φ φ. which implies that B = even and only B = 2, 4 are superficially divergent. → − We now discuss these cases separately. (a) B = 2, D = 2, 2-point function—self energy graph ⇒ Being quadratically divergent, the necessary Taylor expansion for the 2-point function is of the form, 2 2 2 Σ p = Σ (0) + p Σ0 (0) + Σ p
2
where Σ (0) and Σ0 (0) are divergent and Σ p is finite.e Here for conveience we do the Taylor expansion around p2 = 0. To cancel these divergences we need to add two counterterms to the
Lagrangian, e 1 1 Σ (0) φ2 + Σ (0) (∂ φ)2 2 2 0 µ which give the following contributions to the self energy graphs,

Fig 5 Counter term for 2 point function

(b) B = 4, D = 0, 4-point function—vertex graphs ⇒ The Taylor expansion is (4) (4) (4) Γ (pi) = Γ (0) + Γ (pi) where Γ(4) (0) is logarithmically divergent which is to be cancelled by conunterterm of the form e i Γ(4) (0) φ4 4! Fig 6 counter term for 4-point function

To this order the general counterterrm Lagrangian is then

1 2 1 2 i (4) 4 ∆ = Σ (0) φ + Σ0 (0) (∂µφ) + Γ (0) φ L 2 2 4! Note that with the counter terms the Lagrangian is of the form

2 1 2 µ 2 λ 4 1 2 1 2 i (4) 4 + ∆ = (∂µφ) φ φ + Σ (0) φ + Σ0 (0) (∂µφ) + Γ (0) φ L L 2 − 2 − 4! 2 2 4! 1 2 1 2 2 λ (4) 4 = 1 + Σ0 (0) (∂µφ) µ + Σ (0) φ 1 + Γ (0) φ 2 − 2 − 4!   This clearly has the same structure as
the unrenormalized Lagrangian.
More specifically, if we define

1 + Σ0 (0) = Zϕ, φ0 = Zϕφ 2 2 1 µ0 = µ + Σ (0) Zϕ− λ = λ 1 + Γ(4) (0) Z 2 = Z Z 2λ 0
φ− λ φ− then we get
2 1 2 µ0 2 λ0 4 + ∆ = (∂µφ ) φ φ (7) L L 2 0 − 2 0 − 4! 0 which is exactly the unrenormalized Lagrangain. Thus the BPH scheme is equivalent to the conventional substraction scheme but organized differently.

1.2.2 Analysis of divergences The BPH renormalization scheme looks simple. It is remarkable that this simple scheme can serve as the basis for setting up a proof for a certain class of field theory. There are many interesting and useful features in BPH which do not show themselves on the first glance and are very useful in the understanding of this renormalization program. We will now discuss some of them.

(a) Convergence of Feynman diagrams In our analysis so far, we have used the superficial degree of divergences D. It is clear that to 1-loop order that superficial degree of divergence is the same as the real degree of divergence. When we go beyond 1-loop it is possible to have an overall D < 0 while there are real divergences in the subgraphs. The real convergence of a Feynman graph is governed by Weinberg’s theorem ([?]) : The general Feynman integral converges if the superficial degree of divergence of the graph together with the superficial degree of divergence of all subgraphs are negative. To be more explicit, consider a Feynman graph with n external lines and l loops. Introduce a cutoff Λ in the momentum integration to estimate the order of divergence,

Λ (n) 4 4 Γ (p1, , pn 1) = d q1 d qiI (p1, , pn 1; q1, , qi) ··· − ··· ··· − ··· Z0 Take a subset S = q10 , q20 , qm0 of the loop momenta q1, , qi and scale them to infinity and all other momenta fixed.{ Let···D (S)} be the superficial degree{ of··· divergence} associated with integration over this set, i. e., Λ 4 4 D(s) d q0 d q0 I Λ ln Λ 1 ··· m ≤ { } Z0

Figure 1: Fig 7 Divergent 6-point function

where ln Λ is some function of ln Λ. Then the convergent theorem states that the integral over { } q1, , qi converges if the D (S)0 s for all possible choice of S are negative. For example the graph {in the··· following} figureis a 6-point function with D = 2. But the integration inside the box with D = 0 is logarithmically divergent. However, in the BPH− procedure this subdivergence is in fact removed by lower order counter terms as shown below.

Fig 8 Counterterm for 6-point function

This illustrates that a graph with D < 0, may contain subgraphs which are divergent. But in BPH scheme, these divergences are removed by contributions coming from the lower order counterterms. This is why we can focus on those graphs with D 0 and do not have to worry about divergences from the subgraphs. ≥

(b) Primitively divergent graphs A primitively divergent graph has a nonnegative overall superficial degree of divergence but is con- vergent for all subintegrations. Thus these are diagrams in which the only divergences is caused by all of the loop momenta growing large together. This means that when we differentiate with respect to external momenta at least one of the internal loop momenta will have more power in the denomi- nator and will improve the convergence of the diagram. It is then clear that all the divergences can be isolated in the first few terms of the Taylor expansion. In othe words, if the diagrams are not primitively divergent, in the Taylor expansion, the differentiation with respect to external momenta may or may not improve the convergences of the graph.

(c) Disjointed divergent graphs Here the divergent subgraphs are disjointed. For illustration, consider the 2-loop graph given below,

Fig 9 2-loop disjoint divergence

Here we have 2 separate quadratically divergent graphs and is certainly not a primatively divergent graph. It is clear that differentiating with respect to the external momentum will improve only one of the loop integration but not both. As a result, not all divergences in this diagram can be removed by subtracting out the first few terms in the Taylor expansion around external momenta. However, the lower order counter terms in the BPH scheme will come in to save the day. The Feynman integral is written as Γ(4) (p) λ3 [Γ (p)]2 a ∝ with 1 4 1 1 Γ(p) = d l 2 2 2 l µ + iε (l p)2 µ2 + iε Z − − − h (4) i and p = p1 + p2. Since Γ(p) is logarithmically divergent, Γa (p) cannot be made convergent no matter how many derivatives act on it, even though the overall superficial degree of divergence is zero. However, we have the lower order counterm λ2Γ (0) corresponding to the substraction introduced at the 1-loop level. This generates the additional− contributions given in the following diagrams,

Fig 10 2-loop graphs with counterms

which are proportional to λ3Γ (0) Γ (p) . Adding these 3 contributions, we get − λ3 [Γ (p)]2 2λ3Γ (0) Γ (p) − = λ3 [Γ (p) Γ (0)]2 λ3 [Γ (0)]2 − − Since the combination in the first [ ] is finite, the divergence in the last term can be removed by one differentiation. Here we see that··· with the inclusion of lower order counterterms, the divergences take the form of polynomials in external momenta. Thus for graphs with disjointed divergences we need to include the lower order counter terms to remove the divergences by substractions in Taylor expansion.

(d) Nested divergent graphs

(a) In this case, one of a pair of divergent 1PI is entirely contained within the other as shown in the following diagram,

Fig 11 Nested divergence After the subgraph divergence is removed by diagrams with lower order counterterms, the overall divergences is then renormalized by a λ3 counter terms as shown below,

Fig 12 counterterm for nested divergence Again diagrams with lower-order counterterm insertions must be included in order to aggregate the divergences into the form of polynomial in external momenta. (e) Overlapping divergent graphs These diagrams are those divergences which are neither nested nor disjointed. These are most diffi cult to analyze. An example of this is shown below,

Fig 13 Overlapping divergence

The study of how to disentangle these overlapping divergences is beyond the scope of this simple introduction and we refer interested readers to the literature ([?].[?]).

From these discussion, it is clear that BPH renormalization scheme is quite useful in organizing the higher order divergences in a more systematic way for the removing of divergences by constructing the counterterms. In particular, the lower order counter terms play an important role in removing divergences in the subgraphs. The general analysis of the renormalization program has been carried out by Bogoliubov, Parasiuk, Hepp ([?]). The result is known as BPH theorem, which states that for a general renormalizable field theory, to any order in perturbation theory, all divergences are removed by the counterterms corresponding to superficially divergent amplitudes.

1.3 Regularization In carrying out the renormalization, we need first to make divergent integral finite before we can do any mathematically meaningful manipulation. There are two different schemes frequently used to make the integrals finite, Pauli-Villars regularization and dimensional regularization. The latter one is a very powerful method for dealing with theories with symmetries and is used widely in the calculation in gauge theories.

1.3.1 Pauli-Villars Regularization In this scheme, we repalce the propagator by the one where we substract from it another propagator with very large mass,

1 1 1 (µ2 Λ2) 1 ( ) = 0 − for large k k2 µ2 → k2 µ2 − k2 Λ2 (k2 µ2)(k2 Λ2) → k4 − 0 − 0 − − 0 − which will make the integral more convergent. This has the advantage of being covariant as compared to simply cutting off the integral at some large momenta. We will illustrate this by an example of 4-point function from the following graph,

whic gives, ( iλ)2 d4l i i Γ p2 = Γ (s) = − 2 (2π)4 (l p)2 µ2 l2 µ2 Z −
− − With Pauli-Villars regularization this becomes,

2 2 4 2 λ Λ d l 1 Γ p = − 4 2 (2π) 2 2 2 2 2 2 Z (l p) µ (l µ )(l Λ )
− − − − h i Taylor expansion around p2 = 0, gives

Γ p2 = Γ (0) + Γ p2

with e λ2Λ2 d4l 1 Γ (0) = − 2 (2π)4 (l2 µ2)2 (l2 Λ2) Z − − and 2 2 4 2 2 λ Λ d l 2l p p Γ p = − 4 · − 2 (2π) 2 2 2 2 2 2 2 Z (l p) µ (l µ ) (l Λ )
− − − − e h i Since the integral in Γ p2 is convergent enough, we can take the limit Λ2 inside the integral to get, → ∞
2 4 2 e 2 λ d l 2l p p Γ p = 4 · − 2 (2π) 2 2 2 2 2 Z (l p) µ (l µ )
− − − e h i Note that in Γ p2 we have take the limit Λ2 inside the integral because it is a suffi ciently convergent integral. The standard method in the computation→ ∞ of the Feynman graphs is to combine the denominators
by using thee identities,

1 n 1 dz1dz2 dzn = (n 1)! ··· δ 1 zi a a a − (a z + + a z )n − 1 2 n 0 1 1 n n i=1 ! ··· Z ··· X 1 n 1 z1dz1dz2 dzn = n! ··· δ 1 zi a2a a n+1 − 1 2 n 0 (a1z1 + + anzn) i=1 ! ··· Z ··· X Here α1, αn are usually called the Feynman parameters. We will illustrate the derivation of this identity as··· follows. We first verify the simple case of n = 2, ♠

1 1 1 dz1dz2δ(1 z1 z2) dz1 dz1 − −2 = 2 = 2 (a z + a z ) [a1z1 + a2 (1 z1)] [(a1 a2)z1 + a2] Z0 1 1 2 2 Z0 − Z0 − 1 1 1 1 = − = (a1 a2) (a1 a2)z1 + a2 a1a2 −  − 0 Differentiate this relation with respect to a1, 1 1 z1dz1dz2δ(1 z1 z2) 2 = 2 − −3 a a2 (a z + a z ) 1 Z0 1 1 2 2 Then 1 1 1 dz1dz2δ(1 z1 z2) = − −2 a1a2a3 a3 (a z + a z ) Z0 1 1 2 2 1 1 Apply this to the combination 2 and , (a1z1 + a2z2) a3

dz1dz2δ(1 z1 z2) 1 y2dy1dy2δ (1 y1 y2) I = − −2 = 2 − − 3 dz1dz2δ(1 z1 z2) (a z + a z ) a3 [y1a3 + y2 (a1z1 + a2z2)] − − Z 1 1 2 2 Z (1 y1) dy1dz1dz2δ(1 z1 z2) = 2 − − − 3 [y1a3 + (1 y1)(a1z1 + a2z2)] Z − Now define (1 y1) z1 = x1, (1 y1) z2 = x2 − − we get

1 y1 x1 x2 (1 y1) dy1dz1dz2δ( − − − ) 2 − 1 y1 1 I = 2 − 3 [y1a3 + (a1x1 + a2x2)] 1 y1 Z  −  dy1dz1dz2δ (1 y1 x1 x2) = 2 − − − [y a + (a x + a x )]3 Z 1 3 1 1 2 2 Or 1 dz3dz1dz2δ (1 x3 x1 x2) = 2 − − − a a a [z a + (a x + a x )]3 1 2 3 Z 3 3 1 1 2 2 where we have renamed y1 as x3. This verifies the case for combining 3 factors a1a2a3 in the denominator. From this we can see how to verify the most general cases. Using these relations we get, ♠ 1 (1 α) dα = 2 − 3 (l p)2 µ2 (l2 µ2)2 A − − − Z h i where A = (1 α) l2 µ2 + α (l p)2 µ2 = (l αp)2 a2 − − − − − − with
h i a2 = µ2 α (1 α) p2 − − Thus

1 4 2 2 2 d l 2l p p Γ p = λ (1 α) dα 4 · − 3 − (2π) 2 Z0 Z (l αp) a2
− − e 1 4 h 2 i 2 d l (2α 1) p = λ (1 α) dα 4 − 3 0 − (2π) (l2 a2 + iε) Z Z − where we have changed the variable l l + αp and drop the term linear in l. This integral now depends only on l2 and not on any angles. But→ it is more conveient to transform this integral into integration in Euclean space rather than the Minkowski spce. To carry out this, we note that in the complex l0 plane, the integrand has poles at 2 2 l0 = l + a iε ± − h i As shown in the graphs we can rotate the contourp of the integration by 900 by the following procedure (W ick rotation). From Cauchy’stheorem we have

dl0f (l0) = 0 CI where 1 f (l0) = 3 2 2 → 2 2 l0 ( l + a iε) " − q − # 6 Since f (l0) l0− as l0 , the contribution from the circular part of contour C with very large radius vanishes and→ we get → ∞ i ∞ ∞ dl0f (l0) = dl0f (l0) i Z−∞ Z− ∞ Thus the integration path has been rotated from along real axis to imaginary axis (Wick rotation). Chang- ing the variable l0 = il4, so that l4 is real we found

i ∞ ∞ ∞ dl4 dl0f (l0) = i dl4f (il4) = i 3 i − l2 + l2 + l2 + l2 + a2 iε Z− ∞ Z−∞ Z−∞ 1 2 3 4 − 2 2 2 2 2
Define the Euclidean momentum as ki = (l1, l2, l3, l4) with k = l1 + l2 + l3 + l4. The integral is then d4l 1 d4k 1 = i (2π)4 (l2 a2 + iε)3 − (2π)4 (k2 + a2 iε)3 Z − Z − Using spherical coordinates in 4-dimensional Euclidean space, we have

2π π π d4k = ∞ k3dk dφ sin θdθ sin2 χdχ Z Z0 Z0 Z0 Z0 and integrating over angles we found

d4k 1 k3dk 1 = 2π2 ∞ (2π)4 (k2 + a2 iε)3 (2π)4 (k2 + a2 iε)3 Z Z0 − 2 2 − 1 ∞ k dk = 2 3 16π 0 (k2 + a2 iε) Z − Many intergrals in loop integration can be worked out using Gamma and Beta functions. The standard Gamma function is defined by ∞ n 1 u Γ(n) = u − e− du (8) Z0 It follows that ∞ n 1 u ∞ m 1 v Γ(m)Γ(n) = u − e− du v − e− dv Z0 Z0 Now we transform this into 2-dimensional integration by setting u = x2, v = y2,

(x2+y2) 2n 1 2m 1 Γ(m)Γ(n) = 4 dxdye− x − y − Z Using polar coordinates, x = r cos θ, y = sin θ, we get

π/2 r2 2(n+m) 1 2n 1 2m 1 Γ(m)Γ(n) = 4 dre− r − dθ (cos θ) − (sin θ) − Z Z0 π/2 2n 1 2m 1 = 2Γ (m + n) dθ (cos θ) − (sin θ) − Z0 T hus we get the general integrals over trigonometric functions,

π/2 2n 1 2m 1 1 Γ(m)Γ(n) 1 dθ (cos θ) − (sin θ) − = = B (n, m) 2 Γ(m + n) 2 Z0 Or π/2 1 n + 1 m + 1 dθ (cos θ)n (sin θ)m = B , (9) 2 2 2 Z0   where B (n, m) is the usual Beta function. Let u = cos2 θ, we get

1 m 1 n 1 B (n, m) = u − (1 u) − du − Z0 u = x2, 1 2m 1 2 n 1 B (n, m) = 2 x − 1 x − du − Z0 x2
Let t = 1 x2 − tm 1dt Γ(m)Γ(n) B (n, m) = ∞ − = (10) (1 + t)m+n Γ(m + n) Z0 This is the formula used in the computation of the Feynman integral. Using this formula in Eq(10), we get

m 1 t − dt 1 Γ(m)Γ(n m) n = − (t + a2) (a2)n m Γ(n) Z − For our integral the result is

d4k 1 1 = (2π)4 (k2 + a2 iε)3 32π2 (a2 iε) Z − − and iλ2 1 dα (1 α) (2α 1) p2 Γ p2 = − − − 32π2 [µ2 α (1 α) p2 iε] Z0 − − −
It is straightforward to carry oute the integration over the Feynman parameter α to get

1 1 1 2 2 2 2 ˜ 2 ˜ iλ 4µ s (4µ s) 2 ( s ) 2 Γ(p ) = Γ(s) = 2 2 + − ln − 1 − | | 1} for s < 0 32π s (4µ2 s) 2 + ( s ) 2 (  | |  " #) 1 { −1 | | } 2 2 2 2 iλ 4µ s 1 s 2 = 2 2 − tan− for 0 < s < 4µ 32π2 − s 4µ2 s (    −  ) 2 2 1 2 1 iλ s 4µ 1 s 2 (s 4µ ) 2 2 2 = 2 2 + ( − ) ln 1 − − 1 + iπ for s > 4µ 32π ( s "s 2 + (s 4µ2) 2 # ) − Using the same procedure, we can calculate the divergent term to give

iλ2Λ2 α iλ2 Λ2 Γ (0) = dα ln , for large Λ 32π2 α(µ2 Λ2) + Λ2 ' 32π2 µ2 Z − 1.3.2 Dimensional regularization The basic idea here is that since the divergences come from integration of internal momentum in 4- dimensional space, the integral can be made finite in lower dimensional space. We can find a way to define the Feynman integrals as functions of space-time n and carry out the renormalization for lower values of n before analytically continuing in n and taking the limit n 4. We will illustrate this by an example. Consider the integral → d4k 1 1 I = ( )[ ] (2π)4 k2 µ2 (k p)2 µ2 Z − − − which is logarithmically divergent in 4-dimension. If we define this as integration over n dimension − dnk 1 1 I (n) = (2π)4 (k2 µ2) (k p)2 µ2 Z −  − −  then the integral is convergent for n < 4.To define this integral for non-integer values of n, we first combine the denominators using Feynman parameters and make the Wick rotation,

1 dnk I (n) = dα 2 2 Z0 Z (k αp) a2 + iε − − 1 h n i d k 2 2 2 = i dα 2 with a = µ α (1 α) p 0 [k2 + a2 iε] − − Z Z − Now introduce the spherical coordinates in n-dimension and carry out the angular intgrations,

2π π π n ∞ n 1 n 2 d k = k − dk dθ1 sin θ2dθ2 sin − θn 1dθn 1 ··· − − Z Z0 Z0 Z0 Z Z0 n/2 2π ∞ n 1 = n k − dk Γ 0 2 Z   where we have used the formula derived before,

m + 1 √πΓ π 2 sinm θdθ =   m + 2 Z0 Γ 2   where we have used 1 √π Γ = 2 2   The the n dimensional integral is − n/2 1 n 1 2iπ ∞ k − dk I (n) = n dα 2 Γ 0 0 [k2 + a2 iε] 2 Z Z −   The dependence on n is now explicit and the integral is well-defined for 0 < Re(n) < 4. We can extend this domain of analyticity by integration by parts

n 1 1 ∞ k dk 2 ∞ d 1 − = − kndk Γ( n ) [k2 + a2 iε]2 Γ( n + 1) dk [k2 + a2 iε]2 2 Z0 − 2 Z0  −  where we have used zΓ(z) = Γ(z + 1) The integral is now well defined for -2 < Re(n) < 4. Repeat this procedure m times, the analyticity domain is extended to 2m < Re(n) < 4 and eventually to Re(n) . To see what happens as n 4, we can integrate over k−to get → −∞ → n 1 dα I(n) = iπn/2Γ 2 2 2 n/2 − 2 0 [a iε] −   Z − Using the formula, n n Γ 3 2 2 Γ 2 = −n as n 4 − 2 2 2 → 4 n →   −
− we see that the singularity at n = 4 is a simple pole. Expand everything around n = 4, n 2 Γ 2 = + γ + (n 4)B + − 2 4 n E − ··· −  n 4 a − = 1 + (n 4) ln a + − ··· where γ is the Euler constant and B is some constant, we obtain the limit, as n 4 E −→ 2iπ2 1 I(n) iπ2 dα ln[µ2 α(1 α)p2] + iπ2γ −→ 4 n − − − − Z0 and the 1-loop contribution to 4-point function is, λ2 2i 1 Γ(p2) = i dα ln[µ2 α(1 α)p2] + iA 32π2 4 n − − −  − Z0  Taylor expansion around p2 = 0 gives Γ(p2) = Γ(0) Γ(˜ p2) − λ2 2i iλ2 Γ(0) = i ln µ2 + iA 32π2 4 n − ' 16π2(4 n)  −  − and iλ2 1 µ2 α(1 α)p2 Γ(˜ p2) = − dα ln − − 32π2 µ2 Z0   iλ2 1 dα(1 α)(2α 1)p2 = − − − 32π2 [µ2 α(1 α)p2] Z0 − − Clearly this finite part is exactly the same as that given by the method of covariant regulariztion (Pauli- Villars). This method which we substract out first few terms of the Taylor expansion is usually referred to as the momentum subtraction scheme. But in the dimensional regularization seheme all the divergences show up as poles in n = 4. Then we can remove the divergences simply by substracting out the poles at n = 4. This is usually called the minimal subtraction scheme (MS). Usually,these pole at n = 4 are accompanies by terms involoving Euler constant and log (4π). One can substract these terms as well. This is known as the modified minimal subtraction or MS. The 1-loop self energy, which is quadraticlly divergent becomes in dimensional-regularization scheme, λ dnk 1 iλπn/2Γ 1 n iΣ(p2) = = − − 2 − 2 (2π)4 k2 µ2 + iε 32π4(µ2)1 n/2 Z − −
From the relation, n Γ 3 n Γ 1 = − 2 − 2 1 n 2 n   − 2 −
2 we see that the quadratic divergnece has pole at n = 4 and also at n = 2. For n 4 we have,

→ iλµ2 1 iΣ(0) = − 16π2 4 n  −  2 Power counting and Renormalizability

We now discuss the problem of renormalization for more general interactions. It is clear that it is advan- tageous to use the BPH scheme in this discussion.

2.1 Theories with fermions and scalar fields We first study the simple case of fermion ψ and scalar field φ. First consider some simple examples.

(a) 4-fermion interaction where the interaction Lagrangian is given by

_ 2 I = g ψψ L   Let F = number of external fermion lines IF = number of internal fermion lines Then from counting total numbers of fermion lines we have

F + 2 (IF ) = 4n

and from the number of uncontrained momenta

L = (IF ) n + 1 − where n is the number of vertices. The superficial degree of divergence is then

D = 4L (IF ) − L and IF can eliminated to give a simple formula for D, 3 D = 4 F + 2n − 2 This means that for n large enough D > 0 for any value of F and we need infinite number of different type of counterterms. The counter terms needed are _ _ _ _ F = 2 : ψψ, ψ∂ψ,/ ψ∂/∂ψ,/ ψ∂/∂/∂ψ,/ ··· ______F = 4 : ψψψψ, ψ∂ψ/ ψψ, ψ∂ψ/ ψ∂ψ,/ ψψψ∂/∂ψ,/ ψ∂/∂/∂ψψψ,/ ··· ··· Thus we will not be able to aborb these infinities by redefining the parameters in the Lagrangian.

(b) Yukawa interaction with interaction of the form, _ I = f ψγ ψ φ L 5   Then we have F + 2 (IF ) = 2n, B + 2 (IB) = n, and L = (IF ) + (IB) n + 1 − where n is the number of vertices. The superficial degree of divergence is

D = 4L (IF ) 2(IB) − − and can be simplied to give 3 D = 4 B F − − 2 Now D is independent of n. Thus D 0, only for small number of cases and the counterterms needed are ≥ 2 2 4 B = 2, φ , (∂µφ) ,B = 4, φ _ _ _ F = 2; ψψ, ψ∂ψ,/ F = 2,B = 1; φψψ, So the counter terms can be absorbed in the redefinition of the parameters in the Lagrangian.

We now can study the most general cases. Write the Lagrangian density in the general form

= 0 + i L L L i X where 0 is the free Lagrangian quadratic in the fields and i are the interaction terms e.g. L L _ _ 2 _ µ i = g1ψγ ψ∂µφ, g2 ψψ , g3 ψψ φ, L ···     Here ψ denotes a fermion field and φ a scalar field. Define the following quantities

ni = number of i th type vertices − bi = number of scalar lines in i th type vertex − fi = number of fermion lines in i th type vertex − di = number of derivatives in i th type of vertex B = number of external scalar lines− F = number of external fermion lines IB = number of internal scalar lines IF = number of internal fermion lines

Counting the total scalar and fermion lines, we get

B + 2 (IB) = nibi (11) i X

F + 2 (IF ) = nifi (12) i X Using momentum conservation at each vertex we can compute the number of loop integration L as

L = (IB) + (IF ) n + 1, n = ni − i X where the last term is due to the overall momentum conservation which does not contain the loop integra- tions. The superficial degree of divergence is then given by

D = 4L 2 (IB) (IF ) + nidi − − i X Using the relations given in Eqs(11,12) we can write, 3 D = 4 B F + niδi (13) − − 2 i X where 3 δi = bi + fi + di 4 2 − is called the index of divergence of the interaction. Using the fact that Lagrangian density has dimension 3 L 4 and scalar field, fermion field and the derivative have dimensions, 1, , and 1 respectively, we get for the 2 dimension of the coupling constant gi as 3 dim (gi) = 4 bi fi di = δi − − 2 − − Thus the index of divergence of each interaction term is related to the dimension of the corresponding coupling constant. We distinguish 3 different situations;

(a) δi < 0 In this case, D decreases with the number of i-th type of vertices and the interaction is called super renormalizable interaction. The divergences occur only in some lower order diagrams. There− is only one type of theory in this category, namely φ3 interaction.

(b) δi = 0 Here D is independent of the number of i-th type of vertices and interactions are called renormalizable interactions. The divergence are present only in small number Green’s functions. Interactions in _ this category are of the form, gφ4, f ψψ φ.   (c) δi > 0 Then D increases with the number of the i-th type of vertices and all Green’sfunctions are divergent for large enough ni. These are called non renormalizable interactions. There are plenty of examples _ _ 2 − µ 5 in this category, g1ψγ ψ∂µφ, g2 ψψ , g3φ , ···   The index of divergence δi can be related to the operator’s canonical dimension which is defined in terms of the high energy behavior of the propagator in the free field theory. More specifically, for any operator A, we write the 2-point function as

2 4 ip x DA p = d xe− · 0 T (A (x) A (0)) 0 h | | i Z
If the asymptotic behavior is of the form,

2 2 ωA/2 2 DA p p − , as p −→ −→ ∞ then the canonical dimension is defined
as

d (A) = (4 ωA) /2 − Thus for the case of fermion and scalar fields we have,

d (φ) = 1, d (∂nφ) = 1 + n 3 3 d (ψ) = , d (∂nψ) = + n 2 2 Note that in these simple cases, these values are the same as those obtained in the dimensional analysis in the classical theory and sometimes they are also called the naive dimensions. As we will see later for the vector field, the canonical dimension is not necessarily the same as the naive dimension. For composite operators that are polynomials in the scalar or fermion fields it is diffi cult to know their asymptotic behavior. So we define their canonical dimensions as the algebraic sum of their constituent fields. For example, _ d φ2 = 2, d ψψ = 3
  For general composite operators that show up in the those interaction described before, we have, 3 d ( i) = bi + fi + di L 2 and it is related to the index of divergence as

δi = d ( i) 4 L − We see that a dimension 4 interaction is renormalizable and greater than 4 is non-renormalizable. So if we require the theory to be renormalizable, then this will restrict the possible interaction greatly. Basically, _ for theories with scalar and fermion fields, they are of the type φ3, φ4, ψψ φ and their generalization to include internal symmetry indicies.   Counter terms Recall that we add counterterms to cancel all the divergences in Green’s functions with superficial degree of divergences D 0. For convenience we use the Taylor expansion around zero external momenta ≥ pi = 0. It is easy to see that for a general D 0 diagram with given F and B,counter terms will be of the form ≥ α F B Oct = (∂µ) (ψ) (φ) , α = 0, 1, 2, D (14) ··· For example, in the Yukawa theory, the graph with F = 2, 2 fermio lines,has D = 1. So the counter terms required wil have α = 0, 1, and are of the forms, _ _ µ ψψ, ψγµ∂ ψ The the canonical dimension of the counter term in Eq (14) is given by 3 d = F + B + α ct 2 The index of divergence of the counterterms is

δct = dct 4 − Using the relation in Eq (13) we can write this as

δct = (α D) + niδi − i X Since α D , we have the result ≤ δct niδi ≤ i X This means that the counterterms induced by a Feynman diagrams have indices of divergences less or equal to the sum of the indices of divergences of all interactions δi in the diagram. We then get the very useful result that the renormalizable interactions which have δi = 0 will generate counterterms with δct 0. Thus ≤ if all the δi 0 terms are present in the original Lagrangain, then the counter terms will have the same structure as≤ the interactions in the original Lagragian and they may be considered as redefining parameters like masses and coupling constants in the theory. On the other hand non-renormalizable interactions which have δi > 0 will generate counterterms with arbitrary large δct in suffi ciently high orders and clearly cannot be absorbed into the original Lagrangian by a redefinition of parameters δct. Thus non-renormalizable theories will not necessarily be infinite; however the infinite number of counterterms associated with a non-renormalizable interaction will make it lack in predictive power and hence be unattractive, in the framework of perturbation theory. We will adopt a more restricted definition of renormalizability: a Lagrangian is said to be renormalizable by power counting if all the counterterms induced by the renormalization procedure can be absorbed by

redefinitions_ of parameters in the Lagrangian. With this definition the theory with Yukawa interaction ψγ5ψφ by itself, is not renormalizable even though the coupling constant is dimensionless. This is because the 1-loop diagram shown below Fig 14 Box diagram for Yukawa is logarithmically divergent and needs a counter term of the form φ4 which is not present in the original Lagrangain. Thus Yukawa interaction with additional φ4 interaction is renormalizable.

2.2 Theories with vector fields Here we distinguish massless from massive vector fields because their asymptotic behaviors for the free field propagators are very different.

(a) Massless vector field Massless vector field Aµ is usually associated with local gauge invariance as in the case of QED. The asymptotic behavior of free field propagator for such vector field is very similar to that of scalar field. For example, in the Feynman gauge we have

igµν 2 2 ∆µν (k) = − O k− , for large k k2 + iε −→
which has the same asymptotic behaior as that of scalar field. The canonical dimension is then

d (Aµ) = 1

and is the same as the scalar field. Thus the power counting for theories with massless vector field interacting with fermions and scalar fields is the same as before. The renomalizable interactions in this category are of the type, _ µ 2 µ µ ψγµψA , φ AµA , (∂µφ) φA Here Aµ is a massless vector field and ψ a fermion field. Note that in this theory of massless vector field one needs to consider the gauge invariance in the construction♠ of the counter terms. Consider the case of QED where the Lagragian is given by

µ 1 µν = ψ (x) γ (i∂µ eAµ) ψ (x) mψ (x) ψ (x) FµνF L − − − 4 and is invariant under the gauge transformation,

µ µ µ µ ieα A A0 = A ∂ α, ψ (x) ψ0 (x) = e ψ → − → In this theory the superficial degree of divergence is given by 3 D = 4 B F − − 2 Thus for D 0, we have the following cases, ≥ (a) B = 2,F = 0, and D = 2 This means that 1PI 2-point Green’sfunction for the photon field,

4 iq x πµν (q) = d xe− · 0 T (Jµ(x)Jν (0)) 0 h | | i Z is quadratically divergent. Here Jµ(x) is the electromagnetic current and from the current conservation we see that µ ν q πµν (q) = 0, q πµν (q) = 0

This implies that ∆µν (q) has the structure

2 2 πµν (q) = gµνq qµqν π q − µν Thus only ∆ (0) is logarithmically divergent and counterterm

is of the form π (0) FµνF . In µ other words, there is no need for the counter term of the form AµA as a conquence of the gauge invariance. As an example, consider the fermion loop contribution to photon self energy, commonly called the vacuum polarization. From the Feynamn rule we have

d4k 1 1 πµν (q) = tr γµ γν (2π)4 k/ m k/+ q/ m Z  − −  Then 4 µ d k 1 1 q πµν (q) = tr q/ γν (2π)4 k/ m k/+ q/ m Z  − −  We can write q/= (k/+ q/ m) (k/ m) − − − to get 4 µ d k 1 1 q πµν (q) = tr[( )γν] (2π)4 k/ m − k/+ q/ m Z − − If we are allowed to shift the intergration varible k k q in the second term, then we have the cancellation and get the result, → − µ q πµν (q) = 0. But, the integal is highly divergent and the shifting of the integration variable is not allowed. However, if we regulate this integral by dimensional integration, d4k dnk. Then the integral is convergent for suffi ciently small n and we can then shift the integration→ variable to get the result we want. This example illustrate that the relation between vertex function and the propagators, usually referred to as Ward Identity, is indepedent of dimensions and dimensional regulaization is most useful for them. (b) B = 1,F = 2, and D = 0 Here the vertex function is logarithmically divergent and needs a counterterm of the form µ ψ (x) γ Aµψ (x) , which is of the same form as the interaction term in the original Lagrangian. (c) B = 4,F = 0, and D = 0 This seems to give a logarithmically divergent contribution to the light by light scattering graph and would require a counter term of the form A4. Again gauge invariance would come in to save the day. To see this we write the light by light scattering amplitude as

4 4 iqi xi µ ν α β T = d xie− · ε (q1) ε (q2) ε (q3) ε (q4) 0 T (Jµ(x1)Jν(x2)Jα(x3)Jβ(x4)) 0 h | | i Z i=1 µ Y  ν α  β = ε (q1) ε (q2) ε (q3) ε (q4) Tµναβ (q1, q2, q3, q4)

It is easy to see from gauge invariance that

µ ν q Tµναβ (q1, q2, q3, q4) = 0, q Tµναβ (q1, q2, q3, q4) = 0, 1 2 ···

This means that in the Taylor expansion of Tµναβ (q1, q2, q3, q4) the constant term, the term without deivative, vanishes and there is no need for the counterterm. ♠ (b) Massive vector field Here the free Lagrangian is of the form,

1 2 1 2 2 0 = (∂µVν ∂νVµ) + M V L −4 − 2 V µ

where Vµ is a massive vector field and MV is the mass of the vector field. The propagator in momentum space is of the form,

2 i gµν kµkν/MV Dµν (k) = − − O (1) , as k (15) k2 M 2 + iε −→ → ∞ − V
To see this we write the Lagrangian as

4 4 1 2 µ µ ν 2 µ 4 1 2 2 µν µ ν d x 0 = d x Vµ∂ V Vµ∂ ∂ Vν + M VµV = d x Vµ[ ∂ + M g ∂ ∂ ]Vν L 2 − V 2 V − Z Z Z  
The propagator is defined by

2 2 µν µ ν µ 4 [ ∂ + M g ∂ ∂ ]Gνβ(x y) = g δ (x y) V − − β − We solve this equation by the Fourier
transform,

4 ikx Gνβ(x y) = d xe− Dνβ (k) − Z we get 2 2 µν µ ν µ [ k + M g + k k ]Dνβ = g − V β Use the tensor property of Dνβ to write

Dνβ = agνβ + bkβkν

Then 2 2 µ µ 2 2 µ 2 µ µ a k + M g + k kβ + b[ k + M k kβ + k k kβ] = g − V β − V β From these we get h
i
1 1 1 a = , b = −k2 M 2 M 2 k2 M 2 − V V − V Or 2 i gµν kµkν/MV Dµν (k) = − − O(1) as k k2 M 2 + iε −→ → ∞ − V
This implies that canonical dimension of massive vector field is two rather than one,

d(Vµ) = 2

The power counting is now modified with superficial degree of divergence given by 3 D = 4 B F V + ni (∆i 4) − − 2 − − i X with 3 ∆ = b + f + 2v + d i i 2 i i i Here V is the number of external vector lines, vi is the number of vector fields in the ith type of vertex and ∆i is the canonical dimension of the interaction term in . From the formula for ∆i we L see that the only renormalizable interaction involving massive vector field, ∆i 4, is of the form, 2 ≤ φ Aµ and is not Lorentz invariant. Thus there is no nontrivial interaction of the massive vector field which is renormalizabel. However, two important exceptions should be noted; (a) In a gauge theory with spontaneous symmetry breaking, the gauge boson will acquire mass in such a way to preserve the renormalizability of the theory ([?]). (b) A theory with a neutral massive vector boson coupled to a conserved current is also renormaliz- able. Heuristically, we can understand this as follows. The propagator in Eq(15) always appears µ ν 2 between conserved currents J (k) and J (k) and the kµkν/MV term will not contribute because µ µ of current conservation, k Jµ (k) = 0 or in the coordinate space ∂ Jµ (x) = 0. Then the power counting is essentially the same as for the massless vector field case.

2.3 Composite operator In some cases, we need to consider Green’sfunction involving composite operator, an operator with more than one fields at same space time point. We will now illustrate this by an example. Consider a simple 1 2 4 composite operator of the form Ω(x) = 2 φ (x) in λφ theory. Green’s function with one insertion of Ω is of the form, (n) 1 2 G (x; x1, x2, x3, ..., xn) = 0 T ( φ (x)φ(x1)φ(x2)...φ(xn)) 0 Ω | 2 |   In momentum space we have n (n) (n) (2π)4δ4(p+p +p +...+p )G (p; p , p , p , ..., p ) = d4x e ipx d4x e ipixi G (x; x , x , x , ..., x ) 1 2 n φ2 1 2 3 n − i − Ω 1 2 3 n i=1 Z Z Y In perturbation theory, we can use Wick’stheorem to work out these Green’sfunctions in terms of Feynman 1 2 diagram. For example, to lowest order in λ the 2-point function with one composite operator Ω(x) = 2 φ (x) is, after using the Wick’stheorem,

(2) 1 2 G 2 (x; x1, x2) = 0 T φ (x)φ(x1)φ(x2) 0 = i∆(x x1)i∆(x x2) φ 2 | { }| − − or in momentum space G(2)(p; p , p ) = i∆(p )i∆(p + p ) φ2 1 2 1 1 This corresponds to diagram (a) in Fig 15. If we truncate the external propagators, we get

(2) Γ 2 (p, p1, p1 p) = 1 φ − −

Fig 15 Feynman graphs for composite operator To first order in λ, we have

(2) 1 2 ( iλ) 4 4 G 2 (x, x1, x2) = 0 T φ (x)φ(x1)φ(x2) − φ (y) 0 d y φ | {2 4! }| Z   4 iλ 2 = d y − [i∆(x y)] i∆(x1 y)i∆(x2 y) 2 − − − Z The Feynman diagram is given in diagram (b) of Fig 15. The amputated 1PI momentum space Green’s function is 4 (2) iλ d l i i Γ 2 (p; p1, p p1) = − φ − − 2 (2π)4 l2 µ2 + i (l p)2 µ2 + i Z − − − To calculate this type of Green’sfunctions systematically, we can add a term χ(x)Ω(x) to L [χ] = [0] + χ(x)Ω(x) L L where χ(x) is a c-number source function. We can construct the generating functional W [χ] in the presence of this external source. We obtain the connected Green’s function by differentiating ln W [χ] with respect to χ and then setting χ. to zero. Renormalization of composite operators Superficial drgrees of divergence for Green’sfunction with one composite operator is,

DΩ = D + δΩ = D + (dΩ 4) − 1 2 where dΩ is the canonical dimension of Ω. For the case of Ω(x) = φ (x), d 2 = 2 and D 2 = 2 n 2 φ φ − ⇒ only 2-point function Γ(2) is divergent. Taylor expansion takes the form, φ2

Γ(2)(p; p ) = Γ(2)(0, 0) + Γ(2) (p, p ) φ2 1 φ2 φ2R 1 We can combine the counter term i − Γ(2)φ2(0, 0)χ(x)φ2(x) 2 with the original term to write

i i 2 2 i 2 − χφ Γ 2 (0, 0)χφ = Z 2 χφ 2 − 2 φ −2 φ In general, we need to insert counterterm ∆Ω into the original addition

L L + χ(Ω + ∆Ω) → 1 2 If ∆Ω = CΩ, as in the case of Ω = 2 φ ,we have

L[χ] = L[0] + χZΩΩ = L[0] + χΩ0

with Ω0 = ZΩΩ = (1 + C)Ω Such composite operators are said to be mutiplicative renormalizable and Green’s functions of unrenor- malized operator Ω0 is related to that of renormalized operator Ω by

(n) G (x; x1, x2, ...xn) = 0 T Ω0(x)φ(x1)φ(x2)...φ(xn) 0 Ω0 h | { }| i n/2 (n) = ZΩZφ GΩR(x; x1, ...xn) For more general cases, the operator in the counter term might not be the operator itself, ∆Ω = cΩ and the renormalization of a composite operator may require counterterm proportional to other composite6 operators. Example: Conside 2 composite operators A and B. Denote their counterterms by ∆A and ∆B. Including the counter terms with A and B we can write,

L[χ] = L[0] + χA(A + ∆A) + χB(B + ∆B) Very often counterterms ∆A and ∆B are linear combinations of A and B

∆A = CAAA + CABB

∆B = CBAA + CBBB In this case we say there is operator mixing among A and B. Write

A 1 + CAA CAB L[χ] = L[0] + (χA χB) C where C = { } B { } CBA 1 + CBB     Diagonalized C by bi-unitary transformation { } Z 0 U C V + = A0 { } 0 Z  B0  where U, V are 2 2 unitary matrices. Then × 0 0 L[χ] = L[0] + ZA0 χA0 A + ZB0 χB0 B

A0 A = V χ χ = (χA χB) U B0 B A0 B0    
We say that the new combinations A0 ,B0 are multiplicatively renormalizable.

3 Renormalization group

Renormalization scheme requires specification of substraction points which introduce new mass scales. As we will see this introduces the concept of energy dependent "coupling constants",

e.g λ = λ(s) even though the coupling constants in the original Lagranggian are independent of energies.

3.1 Renormalization group equation

In general, there is arbitrariness in choosing the renormalization schemes (or the substraction points). Nevertheless, the physical results should be the same, i.e. independent of renormalization schemes. In essence this is the physical content of the renormalization group equation. Suppose we have different renormalizarion scheme R and R0. From the point of view of BPH renormalization, we can write the bare Lagrangian as, (see Eq(7))

= R(R quantities) = R (R0 quantities) L L − L 0 − Recall that 1 − 2 1 2 2 2 2 φR = ZϕR φ0, λR = ZλR− ZφRλ0 µR = µ0 + δµR Similarly, 1 2 1 2 2 2 2 φ = Z− φ , λR = Z− Z λ0 µ = µ + δµ R0 ϕR0 0 0 λR0 φR0 R0 0 R0

Since φ0, λ0 and µ0 are the same, we can finite relations between R and R0 quantities Callan-Symanzik equation − Here to conform with the standard notation, we make a change of notation. We will use m0 and m for bare and renormalized masses instead of µ0 and µ. The paramter µ is now used to denote the substraction point. In general the renormalized 1PI Green’sfunctions are related to bare 1PI Green’sby

n (n) − 2 (n) Zφ ΓR (Pi, λ, m, µ) = Γ (Pi, λ0, m0)

(n) The renormalized ΓR (Pi, λ, m, µ) depends on the substraction point µ, while the unrenormalied one (n) 2 Γ (Pi, λ0, µ0) does not,

∂ ∂ n (n) µ Γ(n)(P , λ , µ2) = 0, or µ [Z− 2 Γ (P , λ, µ)] = 0 ∂µ i 0 0 ∂µ φ R i Using the µ dependence of Z, λ, m we get

∂ ∂ ∂ (n) µ + β (λ) + m (λ) nγ (λ) Γ (Pi, λ, µ) = 0 ∂µ ∂λ ∂m − R   where ∂λ ∂m 1 ∂ β (λ) = µ , m (λ) = µ , γ (λ) = µ ln Z ∂µ ∂µ 2 ∂µ φ This is usually referred to as Callan-Symanzik equation, or renormalization group equation. Now we want to study the behavior of the Green’s functions as we vary the substraction scale µ. In particular, we want to know the asymptotic behavior at high energies. We will use this parameter µ to scale all other dimensional variable like momenta Pi by defining a dimensionless quantity Γ¯ as P Γ(n)(P , λ, m, µ) = µ4 nΓ¯(n)( i , m, λ) i − R µ For simplicity, we will study the theory with m = 0. Since Γ¯ is dimensionless, as we scale up the momenta Pi,we can write ∂ ∂ σP (µ + σ )Γ¯(n)( i , λ) = 0 ∂µ ∂σ R µ and ∂ ∂ σP [µ + σ + (n 4)]Γ(n)( i , λ) = 0 ∂µ ∂σ − µ From Callan-Symanzik equation we get

∂ ∂ (n) [σ β(λ) + nγ(λ) + (n 4)]Γ (σpi, λ, µ) = 0 ∂σ − ∂λ − To solve this equation, we remove the non-derivative terms by the transformation

λ γ(x) Γ(n)(σp , λ, µ) = σ4 n exp[n dx]Γ(n)(σp , λ, µ) as i − β(x) i Z0 Then Γ(n) satisfies the equation

∂ ∂ (n) [σ β(λ) ]Γ (σpi, λ, µ) = 0 ∂σ − ∂λ or ∂ ∂ (n) t [ β(λ) ]Γ (e pi, λ, µ) = 0 where t = ln σ (16) ∂t − ∂λ Introduce the effective, or running constant λ¯ as solution to the equation dλ¯(t, λ) = β(λ¯) with initial condition λ¯(0, λ) = λ dt This equation has the solution ¯ λ(t,λ) dx t = β(x) Zλ It is straightforward to show that 1 dλ¯ 1 ∂ ∂ = and [ β(λ) ]λ¯(t, λ) = 0 β(λ¯) dλ β(λ) ∂t − ∂λ

This means that any funciton F (n) which depends on t and λ through the combination λ¯(t, λ) will satisfy the renormalization equation in Eq(16)

∂ ∂ (n) [ β(λ) ]F (pi, λ¯(t, λ), µ) = 0 ∂t − ∂λ For the other factor of the renormalization group equation, we can write it in terms of effective coupling constant as ¯ λ γ(λ) λ γ(x) λ γ(x) exp[n dλ] exp[n dx + n dx] 0 β(λ) ∼ 0 β(x) λ¯ β(x) Z Z ¯ Z λ γ(x) = H(λ¯) exp[ n dx] − β(x) Zλ where ¯ λ γ(x) H(λ¯) = exp[n dx] β(x) Z0 The solution is then t (n) 4 n (n) Γ (σpi, λ, µ) = σ − exp[ n γ(λ¯(x0, λ))dx0]H(λ¯)F (pi, λ¯(t, λ), µ) as − Z0 If we set t = 0 (or σ = 0), we see that

(n) (n) Γas (pi, λ, µ) = H(λ)F (pi, λ, µ) Thus the solution has the simple form t (n) 4 n (n) Γ (σpi, λ, µ) = σ − exp[ n γ(λ¯(x0, λ))dx0 ]Γ (pi, λ¯(t, λ), µ) as − as Z0 4 n We now discuss the properties of this solution.The first factor σ − comes from the naive dimension of the (n) Green’s function Γas (pi, λ¯(t, λ), µ) as we scale up the momenta by increasing σ. The second factor gives the dependence on t = ln σ apart from the navie diemension and is usually called the anomalous dimension. The last factor is the original unscaled Green’s function but with the coupling constant replaced by the effective coupling constant λ¯(t, λ).

3.2 Effective coupling constant We now study the asymptotic behavior of the effective coupling constant λ¯(t, λ) in order to understand the asymptotic behavior of the Green’sfunctions. The differential equation goverens the behavior of λ¯(t, λ) is a first order equation and can be studies as follows. dλ¯(t, λ) = β(λ¯) initial condition λ¯(0, λ) = λ (17) dt Consider the case where β(λ) has the following simple behavior

Suppose 0 < λ < λ1, then from Eq(17) we see that, dλ¯ t=0> 0 dt | dλ¯ λ¯ increases as t increases. This increase will continue until λ¯ reaches λ1, where = 0 ⇒ dt On the other hand, if initially λ1 < λ < λ2, then dλ¯ t=0< 0 dt | λ¯ will decrease until it reaches λ1. Thus as t , we get → ∞

lim λ¯(t, λ) = λ1 λ1 : ultraviolet stable fixed point t →∞ and (n) ¯ (n) Γas (pi, λ(t, λ), µ) t Γas (pi, λ1, µ) → →∞ Example: Suppose β(x) has a simple zero at λ = λ1,

β(λ) a(λ1 λ) a > 0 ' − Then ¯ dλ at = a(λ1 λ) λ¯ = λ1 + (λ λ1)e− dt − ⇒ − i.e. the approach to fixed point is exponential in t, or power in t = ln σ. Also the prefactor can be simplified,

t λ¯ λ¯ γ(y)dy γ(λ1) dλ0 γ(λ1) λ¯ λ1 γ(λ¯(x, λ))dx = − = − ln( − ) β(y) ≈ a λ0 λ1 a λ λ1 Z0 Zλ Zλ − − = γ(λ1)t = γ(λ1) ln σ

(n) 4 n[1+γ(λ1)] (n) lim Γ (σpi, λ, µ) = σ − Γ (pi, λ1, µ) σ as as →∞ Thus the asymptotic behavior in field theory is controlled by the fixed point λ1 and γ(λ1) anomalous dimension.

(a) Asymptotic Freedom

I t is the zeros of the β function of the effective coupling constant λ¯(t, λ) which determines the aymp- totic behavior of the Green’sfunctions.− But the locations of the zeros of β function need nonperturbative computation and is beyond reach in most cases. However, pertuabation theory− can tell us the behavior of the β function near the origin of the coupling constant space. As we will the discuss later in Chapter on QCD,− to explain the data from deep-inelastic electron-proton scattering, it require some theory which behaves like free field theory in the high energy regime. Thie is usually referred to as asymptotic freedom. We now examine the β function for each of the renomalizable theory we have studied. − (a) λφ4 theory The Lagrangian is 1 2 2 2 λ 4 = [(∂µφ) m φ ] φ L 2 − − 4! Effective coupling constant λ¯ to lowest order satisfies the differential equation

dλ¯ 3λ2 = β(λ¯) , β(λ) + O(λ3) dt ≈ 16π2 It is not asymptotically free. The generalization to more than one scalar fields is the replacement,

4 λφ λijklφ φ φ φ , λijkl is totally symmetric → i j k l Then the differential equations are of the form, dλ 1 β = ijkl = [λ λ + λ λ + λ λ ] ijkl dt 16π2 ijmn mnkl ikmn mnjl ilmn mnjk

3 For the special case, i = j = k = l = 1, we wee that β1111 = 16π2 λiimnλmn11 > 0 and theory is not as- ymptotically free. (b) Yukawa interaction Here we need to include the scalar self interaction λφ4 in order to be renormalizable

µ 1 2 2 2 4 = ψ¯(iγ ∂µ m)ψ + [(∂µφ) µ φ ] λφ + fψψφ¯ L − 2 − − Now we have a coupled differential equations, dλ β = = Aλ2 + Bλf 2 + Cf 4, A > 0 λ dt df β = = Df 3 + Eλ2f, D > 0 f dt 2 To get βλ < 0 , with A > 0 , we need f λ. This means we can drop E term in βf . With D > 0, Yukawa coupling f is not asymptotically free.∼ Generalization to the cases of more than one fermion fields or more scalar fields will not change the situation. (c) Abelian gauge theory(QED) The Lagrangian is of the usual form,

µ 1 µν = ψiγ¯ (∂µ ieAµ)ψ mψψ¯ FµνF L − − − 4 _ The effective coupling constant e satisfies the equation, _ _ de e3 = β = + O(e5) dt e 12π2 For the scalar QED we have _ _ de e3 = β = + O(e5) dt e0 48π2 Both are not asymptotically free. (d) Non-Abelian gauge theories It turns out that only non-Abelian gauge theories are asymptotically free. Write the Lagrangian as

1 µν = Tr(FµνF ) L −2 where a Fµν = ∂µAν ∂νAµ ig[Aµ,Aν],Aµ = TaA − − µ and 1 [T ,T ] = if T ,T (T ,T ) = δ a b abc c r a b 2 ab The evolution of the effective coupling constant is governed by dg g3 11 = β(g) = ( )t2(V ) < 0 dt −16π2 3 Since β(g) < 0 for small g, this theory is asymptotically free. Here ab t2(V )δ = tr[Ta(V )Tb(V )] t2(V ) = n for SU(n) If gauge fields couple to fermions and scalars with representation matrices, T a(F ) and T a(s) respec- tively, then g3 11 4 1 β = t2(V ) + t2(F ) + t2(s) g 16π2 − 3 3 3   where ab a b t2(F )δ = tr(T (F )T (F )) ab a b t2(S)δ = tr(T (S)T (S)) 4 Appendix n-dimensional Integration

4.1 n-dimensional “spherical” coordinates In a n-dimensional space, the Cartesian coordinates can be parametrized in terms of the “spherical” coordinates as

x1 = rn sin θn 1 sin θn 2.... sin θ2 sin θ1, − − x2 = rn sin θn 1 sin θn 2.... sin θ2 cos θ1, − − x3 = rn sin θn 1 sin θn 2.... sin θ3 cos θ2, − − ......

xn = rn cos θn 1 (18) − where 0 θ1 2π, 0 θ2, θ3....θn 1 π, (19) ≤ ≤ ≤ − ≤ and 2 2 2 2 rn = x1 + x2 + ... + xn (20) We want to show that the n-dimensional infinitesimal volume is given by

dx1dx2dx3...dxn (21) n 1 n 2 n 3 = rn− (sin θn 1) − (sin θn 2) − .... (sin θ2)(dθ1dθ2....dθn 1) drn − − − We will do this by finding the relation between the volume factors in n and n 1 dimensions. Namely, we will proceed from the simplest n = 2 to higher and higher dimensional cases: −

(n = 2) (n = 3) (n = 4) (general n) (22) −→ −→ −→ 4.1.1 (a) n = 2

Here the two Cartesian coordinates (x1, x2) are related to the familiar polar coordinates (r2, θ1),

2 2 2 x1 = r2 sin θ1, x2 = r2 cos θ1, 0 θ1 2π, r = x + x (23) ≤ ≤ 2 1 2

The distance ds2 between neighboring points is given by

2 2 2 2 2 2 (ds2) = (dx1) + (dx2) = (dr2) + r2 (dθ1) (24)

The “volume”element is then the product of segments in orthogonal directions – i.e. the product of the coordinate differential with the appropriate coeffi cients as indicated by the (quadratic) distance relation:

dV2 = dx1dx2 = (dr2)(r2dθ1) = r2dr2dθ1. (25)

4.1.2 (b) n = 3

Consider a sphere in three-dimensions. If we cut this sphere by a plane perpendicular to x3-axis, we get a series of circles in the planes spanned by Cartesian coordinates (x1, x2) which are related to the polar coordinates (r2, θ1)

2 2 2 x1 = r2 sin θ1, x2 = r2 cos θ1, 0 θ1 2π r = x + x (26) ≤ ≤ 2 1 2 The infinitesimal distance on this plane can be expressed in these two coordinate systems as

2 2 2 2 2 (dx1) + (dx2) = (dr2) + r2 (dθ1) (27) We can also cut the sphere by a plane containing the x3 axis, resulting in a series of “vertical circles”. On these two dimensional subspaces, the Cartesian coordinates are (r2, x3) and the corresponding polar coordinates are (r3,θ2) ..We recognize that θ2 is the usual polar angle.

r2 = r3 sin θ2, x3 = r3 cos θ2, 0 θ2 π (28) 2 2 2 2 2 2 ≤ ≤ r3 = r2 + x3 = x1 + x2 + x3, (29)

Just like eqn (30), the infinitesimal distance can be expressed in two equivalent ways:

2 2 2 2 2 (dr2) + (dx3) = (dr3) + r3 (dθ2) (30)

Combining the two sets of coordinates in eqns (26) and (28), we get the usual spherical coordinate relations,

x1 = r3 sin θ2 sin θ1

x2 = r3 sin θ2 cos θ1 (31)

x3 = r3 cos θ2

We can turn the distance formula,

2 2 2 2 (ds3) = (dx1) + (dx2) + (dx3) , into 2 2 2 2 2 (ds3) = (dr2) + r2 (dθ1) + (dx3) (32) by using eqns (27). This can be further reduced, by eqn (30) and (28), into

2 2 2 2 (ds3) = (dx1) + (dx2) + (dx3) 2 2 2 2 2 = (dr3) + r2 (dθ1) + r3 (dθ2) 2 2 2 2 2 2 = (dr3) + r3 sin θ2 (dθ1) + r3 (dθ2) (33)

The volume element can be obtained from the product of these three terms

2 dV3 = (dr3)(r3 sin θ2dθ1)(r3dθ2) = r3 sin θ2 (dr3dθ1dθ2) . (34)

4.1.3 (c) n = 4

We can also imagine cutting the sphere in four-dimensions by the manifold (x4 = constant) to get 3-sphere with radius, r3 = r4 sin θ3, where we can introduce three-dimensional spherical coordinates, (r3, θ1, θ2) as in eqn (31)

x1 = r3 sin θ2 sin θ1 = (r4 sin θ3) sin θ2 sin θ1

x2 = r3 sin θ2 cos θ1 = (r4 sin θ3) sin θ2 cos θ1 (35)

x3 = r3 cos θ2 = (r4 sin θ3) cos θ2 with the distance formula (33)

2 2 2 2 2 2 2 2 2 (dx1) + (dx2) + (dx3) = (dr3) + r3 (dθ2) + r3 sin θ2 (dθ1) (36)

Now we introduce two-dimensional polar coordinates (r4, θ3) in the (r3, x4) plane

r3 = r4 sin θ3, x4 = r4 cos θ3 (37) with the distance formula 2 2 2 2 2 (dr3) + (dx4) = (dr4) + r4 (dθ3) (38) In this way the infinitesimal distance in this four dimensional space can be rewritten by using eqns (36) and (38)

2 2 2 2 2 (ds4) = (dx1) + (dx2) + (dx3) + (dx4) (39)

h 2 2 2 2 i2 2 2 = (dr3) + r3 (dθ2) + r3 sin θ2 (dθ1) + (dx4)

h 2 2 2 2 2 i2 2 = (dr3) + (dx4) + r3 (dθ2) + r3 sin θ2 (dθ1) h 2 2 i2 2 2 2 2 2 = (dr4) + r4 (dθ3) + r3 (dθ2) + r3 sin θ2 (dθ1)

The infinitesimal volume element is then

dv4 = (dr4)(r4dθ3)(r3dθ2)(r3 sin θ2dθ1) 3 2 = r4 sin θ3 sin θ2 (dr4dθ1dθ2dθ3) (40)

where we have used eqn (37) to reach the last expression.

4.1.4 (d) General n

x1 = rn sin θn 1 sin θn 2.... sin θ2 sin θ1, − − x2 = rn sin θn 1 sin θn 2.... sin θ2 cos θ1, − − x3 = rn sin θn 1 sin θn 2.... sin θ3 cos θ2, − − ......

xn = rn cos θn 1 (41) − where 2 2 2 2 rn = x1 + x2 + .... + xn, 0 θ1 2π, 0 θ2, θ3...θn 1 π (42) ≤ ≤ ≤ − ≤ The infinitesimal distance is

2 2 2 2 (dsn) = (dx1) + (dx2) + ..... + (dxn) (43) 2 2 2 2 2 2 = (drn) + rn (dθn 1) + rn sin θn 1 (dθn 2) − − − 2 2 2 2 +rn sin θn 1 sin θn 2 (dθn 3) + ...... − − − 2 2 2 2 2 +rn sin θn 1 sin θn 2...... sin θ2 (dθ1) − − and the volume element is

dVn = dx1dx2.....dxn (44) n 1 n 2 n 3 = (rn) − (sin θn 1) − (sin θn 2) − .... sin θ2 (drndθ1dθ2....dθn 1) − − − 4.2 Some integrals in dimensional regularization Use the dimensional regularization we can derive the following results for the Feynman integrals with denominator power α in n dimensions:

(a)

dnk 1 I (α, n) = 0 (2π)n (k2 + 2p k + M 2 + iε)α Z n · 2 n ( π) Γ α 2 1 = i − n − n (45) 2 2 α (2π) Γ(α) (M p + iε) − 2
− (b) dnk k I (α, n) = µ µ (2π)n (k2 + 2p k + M 2 + iε)α Z · = pµI0 (α, n) (46) − (c) dnk k k I (α, n) = µ ν µν (2π)n (k2 + 2p k + M 2 + iε)α Z · 1 M 2 p2 = I0 (α, n) pµpν + gµν − (47) 2 α n 1 " − 2 − #
dnk k k k I (α, n) = µ ν ρ µνρ (2π)n (k2 + 2p k + M 2 + iε)α Z · 1 M 2 p2 = I0 (α, n) pµpνpρ + (gµνpρ + gµρpν + gνρpµ) − 2 α n 1 " − 2 − # We will do them in sequence.

(a) In the Feynman integral dnk 1 I (α, n) = , (48) 0 (2π)n (k2 + 2p k + M 2 + iε)α Z · the denominator can be written as

D = k2 + 2p k + M 2 + iε = (p + k)2 + M 2 p2 + iε 2 · 2 2 − = k0 + M p + iε (49) −
where k = k + p. For the case p2 M 2, we
can perform the Wick rotation to get 0 ≥ 2 D = k + M 2 p2 + iε = (k¯2 + a2) (50) − − − where a2 = p2 M 2 i and − − dnk 1 1 I (α, n) = i . (51) 0 (2π)n ( 1)α (k2 + a2)α Z − As usual [see CL-eqn (2.112)], the n-dimensional angular integration gives,

n 2π 2 dΩn = (52) Γ n Z 2 Then
α n n 1 i ( 1) 2π 2 ∞ k − dk I0 (α, n) = − (2π)n Γ n (k2 + a2)α 2 Z0 α n n 1 i ( 1) 2π 2 1 ∞ t 2 − dt = −
(2π)n Γ n 2 (t + a2)α 2 Z0 α n n n i ( 1) π2
1 Γ 2 Γ α 2 = − n n α n − (2π) Γ 2 2 Γ(α) 2 (a ) −

n 2 n ( π) Γ α
2 1 = − n − n (53) 2 2 α (2π) Γ(α) (M p + iε) − 2
− One of the most common convergent Feynman integral has α = 3, dnk 1 I0 (3, n) = n (2π) (k2 + 2p k + M 2 + iε)3 Z n · 2 n ( π) Γ 3 2 1 = − n − n (54) 2 2 3 (2π) Γ (3) (M p + iε) − 2
− which gives for n = 4,

d4k 1 I0 (3, 4) = (2π)4 (k2 + 2p k + M 2 + iε)3 Z · i 1 = (55) 32π2 (M 2 p2 + iε) − (b) dnk k I (α, n) = µ (56) µ (2π)n (k2 + 2p k + M 2 + iε)α Z · As before, we set k0 = k + p. Then

n d k k0 pµ I (α, n) = 0 µ − (57) µ (2π)n (k2 + a2)α Z

and the term linear in kµ0 gives zero because of the symmetric integration. The result is

Iµ (α, n) = pµI0 (α, n) (58) − (c)

dnk k k I (α, n) = µ ν (59) µν (2π)n (k2 + 2p k + M 2 + iε)α Z · n d k k0 pµ (k0 pν) = µ − ν − (2π)n (k2 + a2)α Z
dnk k k dnk 1 = µ ν + p p (2π)n (k2 + a2)α µ ν (2π)n (k2 + a2)α Z Z 1 2 In the first term we can replace kµkν k gµν to get, → n dnk k k g dnk k2 µ ν = µν (60) (2π)n (k2 + a2)α n (2π)n (k2 + a2)α Z Z n 2 2 2 gµν d k k + a a = − n (2π)n (k2 + a2)α Z gµν 2 = I0 (α 1, n) a I0 (α, n) n − −   Using the identity, Γ(x + 1) = xΓ(x) ,we get,

(α 1) a2 I0 (α 1, n) = − I0 (α, n) (61) − α 1 n − − 2 and
1 2 2 1 Iµν (α, n) = pµpν + gµν[M p ] I0 (α, n) (62) 2 − α n 1 " − 2 − #
For the case α = 4, we have

n d k kµkν Iµν (α, n) = n (63) (2π) (k2 + 2p k + M 2 + iε)4 Z · 1 2 2 1 = pµpν + gµν[M p ] I0 (4, n) 2 − 3 n " − 2 # which gives for n=4,

1 2 2 i 1 Iµν (4, 4) = pµpν + gµν[M p ] (64) 2 − 96π2 (M 2 p2 + iε)   − (d)

dnk k k k I (α, n) = µ ν ρ µνρ (2π)n (k2 + 2p k + M 2 + iε)α Z · n d k k0 pµ (k0 pν) k0 pρ = µ − ν − ρ − (65) (2π)n (k2 + a2)α Z

Dropping terms with odd powers of k0, we get dnk 1 Iµνρ (α, n) = [ (kµkνpρ + kµpνkρ + pµkνkρ) pµpνpρ] (2π)n (k2 + a2)α − − Z = pµpνpρI0 (α, n) (pρIµν + pµIνρ + pνIµρ) − − 1 M 2 p2 = I0 (α, n) 4pµpνpρ + (gµνpρ + gµρpν + gνρpµ) − 2 α n 1 " − 2 − #